In the low-dimensional setting, FS and MIO were tuned over \(k=0,\ldots ,20\) . For the high dimensional setting,
FS was tuned over \(k=0,\ldots ,50\) , while MIO was tuned over \(k=0,\ldots ,30\)  only due to time computational constraints. We re-observed that MIO based on the Gurobi optimizer is quite time consuming for high dimensional cases (see [1]}). In this simulation we ran MIO through the R package bestsubest offered in [2]} and we fixed the default budget to 1 minute 40 seconds (per problem per \(k\) ).
